We have proposed a certain modification of the valley approach for the calculation of the instanton-induced cross sections which has the advantage of being formulated in Minkowski space. Namely, we work out a systematic technique to do the semiclassical expansion around certain classical field trajectories in the double functional integral. From the viewpoint of the standard technique [191for
1.1. Balitsky et a!. / Instanton-induced cross sections 77
the expansion around pure instanton fields, what we actually propose is to make one more shift of variables in the double functional integral, such that large 1/g fields of the produced W-bosons are simulated by a classical field with particular analytic properties which we call the “alien” field. By our procedure we automati-cally pick up the proper cut of the forward scattering amplitude in fig. 1, corresponding to processes with baryon number violation. This separation becomes important to c10’~’3 accuracy, from which the “hard—hard” quantum corrections come into the game [16].
The contribution to the function F(E) (1.2) of the order of c8~still can be evaluated by the analytical continuation of the euclidean functional integral. It is known [30] that the sum of soft—soft corrections depends to this accuracy on the choice of constraints in the gauge propagator at the one-instanton background and this dependence should be remedied by soft—hard corrections. Similarly, in the valley approach the semiclassical answer depends to this accuracy on the particular choice of the valley. This ambiguity can most easily be demonstrated by a direct calculation, choosing a different quantum mechanical valley field instead of the simplest kink—antikink configuration in (3.9), and retaining the conformal ansatz.
Namely, we remind of an old result [10], that the field
~,(t,a)=~m[tanh(icm(t+a))—tanh(’~m(t—a))]+0(1/~~),
m= 1_6/~2+O(1/~4), ~‘=2cosh ma (5.1)
provides an approximate solution of the valley equation with the weight function
to(t, a) = 1:
~= _L(~)~+O(1/~2). (5.2)
We evaluate the action on the QCD valley corresponding to (5.1). A simple calculation yields
- 16ir2 6 42
g2 (i_~_~+...). (5.3)
(Note that the accuracy of (5.1) is enough to get the third term.) The 0(1/~~) contribution to the action of the modified valley in (5.3) clearly deviates from the corresponding term in (3.19), as expected. One can easily convince oneself that the saddle point values of collective coordinate are left intact by this modification of the valley, and hence the corresponding contribution to F(c) is easily calculated by the insertion of the value of ~ corresponding to (4.16). It is worth while to remember that variations of the valley > O(1/~2) are forbidden, since it must coincide to O(1/~2) accuracy to the negative mode, see sect. 3. Thus, quite
78 1.1. Ba!itsky et a!. /!nstanton-ind~cedcross sections
generally, the action SV depends on the choice of the valley to the accuracy
~ and yields different contributions to the function F(c) (1.2) to the order of Since all the freedom in the valley equation (3.3) is due actually to a variety of possibilities to insert the unit factor (3.6) in the functional integral, it is clear that the complete answer cannot depend on the choice of the valley. If the integration near the valley is done rigorously, i.e. if all “quantum” corrections to the semiclas-sical result are taken into account, then the answer must become unambiguous. In the particular case of the O(c8’~3)contribution this means that the difference must be compensated by the corresponding change of the soft-hard corrections or by the change of the I-Iiggs component of the valley, which we do not consider in this paper. An interesting question is whether the valley field can be chosen in such a way that all soft-hard corrections vanish to the required 0(1/~4) accuracy (in the exponent). The finding of such an “improved” valley would mean that it is possible to treat soft—hard corrections semiclassically, which is not trivial. This question is under study and we plan to discuss it in a separate publication.
The authors are grateful to D.I. Diakonov, S.Yu, Khlebnikov, A.H. Mueller, V.V. Khoze, V. Yu. Petrov, A. Ringwald and V.1. Zakharov for interesting discussions. In the process of preparing the manuscript we have received a preprint by Arnold and Mattis [32] covering similar topics, and acknowledge the overlap with some of their results. One of us (1.B.) would like to thank the theory groups at CERN and Heidelberg for their hospitality. His stay in Germany was supported in part by the Deutsche Forschungsgemeinschaft.
Appendix A. Gauge phase factors at the background of the valley
The straight-line ordered phase factor
[x+Ae, x+ccer=P ex~{i~idteaA~(x+Ie)} (A.1) is determined as the solution of the differential equation
~[x5,d x+cce]’=ige~A~(x5)[x5,x+ccef (A.2)
(whereX5 = x+Ac) with the boundary condition
[x4, x+cce]v—s 1. (A.3)
The structure of singularities of the gauge factor (Al) follows that of the background field. Thus it is sufficient to work out the formulas for one particular case, say, for the ii valley with the Feynman prescription of encircling the
1.1. Bal0sky et a!, / Instanton-induced cross sections 79
singularities. Going over to the pair of “native” and “alien” fields one only needs to change the prescriptions to those in eqs. (4.1)—(4.6). Therefore, in what follows we drop all the necessary Ic terms which are easily recovered in the final expressions.
We remind that e= (1, 0, 0, 0) is the unity vector in the time direction.
Unfortunately, we have not succeeded in obtaining an analytical solution of (A.2) at the background of an II valley configuration with an arbitrary Ii separation R.
However, to the semiclassical accuracy it is sufficient to evaluate the gauge factors at the saddle-point configuration in eq. (4.16) with the saddle-point value of R collinear to the time axis. In this case the solution of eq. (A.2) is easily found to be
1 RJ~-(Rr) I R~-(Rx) 1
[x5 x+ ccc] = — I + + — 1— —~- (A.4)
2 (Rx)~~ 2 (Rx)%1~ 1’F4
where
F - (R~)+(~v)~’~~ (R,x5-R)-(R~)~ ~
A (R)-(Rr)~ (Rx4-R)+(Rx)v’~
(R, x~-a) -(R~)V~ (R, XAb) +(Rv)V~
x
(A.5)(Rx4-a)+(Rx)V~ (~4-b)-(P~v)V~
and
R2x2
Kl (Rx~
2/ 7 2
R ~ Ptci
= 1 — (~)2 (A.6)
By a staightforward calculation one obtains
dF4 1 1 1 1
= 2(R~)V~— 2 2 + 2 + 2 — 2 FA,
dA XA~P2 (xA—R) —p~ (xA—a) (x5—b)
(A.7) from which we get
I Ri~—(R~)I dF
IcaA’(xa A)=—2 (Rx)~/~ FA dA—~ ‘ (A8) Using eqs. (A.7) and (A.8) it is easy to check that the gauge factor in (A.4) satisfies eq. (A.2).
80 1.1. Ba!itsky ci a!./ Instanton -induced cross sections
As mentioned in the main text, in order to find the amplitude for the emission of real W’s with virtuality p2—s 0, one should pick up the contribution of the region in the coordinate space corresponding to the “Bjorken limit” x2 —s cc,
(xR)—s cc, (xR)/x2 1. In this limit the function F5 simplifies to
2 2
(xA—R) (x5—a)
FA = 22 (A.9)
(x5—b) x5
and the expression for the gauge factor in (A.4) reduces to (4.10) where the prescriptions to go around the singularities are shown explicitly. The expression for the valley field in the temporal gauge is easily obtained using the representation in (4.9). The only nontrivial point is that simplification of the expression in (A.5) to the one in (A.9) can only be done after the differentiation with respect to Xa~since e.g. e2x2 <<(ex)2, but 2e2xa 2(ex)ea. The answer is given in eq. (4.11).
It is instructive to demonstrate that the observed cancellation of additional gauge singularities is a general effect which does not depend on the particular expression for the valley field in eqs. (3.20)—(3.24). To this end we consider the generalized expression for the valley in (3.16) with an arbitrary profile function 4 (going to unity in both limits x2 —s0 and x2 —s cc to ensure zero topological charge of the valley field):
A~(x)=__[ua~_Xa]X2cIi ln —f— . (A.l0) Making the shift x—sx—x0 and the inversion with respect to the point aa:
(x— a)a —sr2/(x —a)2(x — a)a we obtain the generalized valley field in the form i (x-a)~(x-b)(b-ii) (xb)a give the necessary formulas relating the parameters x0, a, r2 of the conformal transformation to the final parameters x1 = R, x2= 0, Pt’ p2 of the II configura-tion,
I.L Ba!iisky ci al. /Jnstanton-induced cross sections 81
r
Pt
r 2p~
P2 2 2 (A.12)
—C
Note that we use the minkowskian notations here in contrast to sect. 3.
It can be demonstrated that the expression for the gauge factor in (A.4) holds true with the substitution of (A.5) by
1 1 (x,_b)2c2
F5=exp 2(Rx)Y~f dl 2 2 ~ lii — 2 2
(x,—a) (x,—b) (x1—a) p
(A.13) which simplifies in the required limit x2 —s cc,(xR) —scc,(xR)/x2 1 to
FA= exp
{
fln(ln(—c2/p2)— 5—h)2c2/(~A _a)202)d~(
T)). (A.14)Proceeding in the same way as above one obtains the expression for the valley field in the temporal gauge in the form
R~uRxR (x b—a)
= 8(Rx)3 {F~1_1+2 (x —b)2
R2x~Rf 1 (x,b—a) 1
+ a ——1—2 4— (A.15)
8(Rx)3 F1) (x—a)2 F11
where the argument of the function ~ is the same as in (A.I1), and F0 = F5~1.It is seen now that for x—sa and x—, h the upper limit of the integration in (A.14) tends to +cc, —cc, respectively. Since 4(s-)—s 1 in both limits, we obtain
x—‘a
F
(x_a)2
x—’b 2
F—~ (x—b) . (A.16)
From (A.16) it easily follows that both expressions in the braces in (A.15) are nonsingular for x—sa and x —sb. It can also be shown that the Fourier transform of (A.I4) is free of the unphysical poles I/p4.
82 1.1. Ba!itsky et a!./ Instanton -induced cross sections
References
[1] G. ‘t Hooft, Phys. Rev. D14 (1976) 3432 [2] A. Ringwald, Nuel. Phys. B330 (1990) 1
[310. Espinosa. NucI. Phys. B343 (1990) 310
[4] L. McLerran, A. Vainshtein and M. Voloshin, Phys. Rev. D42 (1990) 171 [5] VI. Zakharov. NucI. Phys. B353 (1991) 683
[6] J.M. Cornwall, Phys. Len. B243 (1990) 271;
H. Goldberg, Phys. Lett. B246 (1990) 445
[7] T. Banks, G. Farrar, M. Dine and B. Sakita, Rutgers preprint RU-90-20 (1990) [8] V.1. Zakharov, preprint TPI-MINN-90/7-T (1990); NucI. Phys. B371 (1992) 637 [9] DI. Dyakonov and V. Petrov, NucI. Phys. B245 (1984) 259:
E.V. Shuryak, Nuel. Phys. B302 (1988) 559, 574, 599, 621 [10] LI. Balitsky and A.V. Yung, Phys. Lett. Bl68 (1986) 113 [11] A.V. Yung, NucI. Phys. B297 (1988)47
[12] J. Verbaarschot, NucI. Phys. B362 (1991) 33
[13] V.V. Khoze and A. Ringwald. NucI. Phys. B355 (1991) 351
[141 DI. Dyakonov and V. Petrov, Proc. XXYI LNPI Winter School, Leningrad, 1991, p. 8 [15] PB. Arnold and M.P. Mattis, Mod. Phys. Lett. A6 (1991) 2059
[16] A.H. Mueller, Nucl. Phys. B364 (1991) 109
[17] VI. Zakharov, preprint MPI-PAE/PThII/91 (1991); Nucl. Phys. B353 (1991) 683 [18] M. Maggiore and M. Shifman, Mud. Phys. B365 (1991) 161
[19] S.Yu. Khlebnikov, V.A. Rubakov and PG. Tinyakov, NucI. Phys. B350 (1991) 451 [20] I.!. Balitsky and V.M. Braun, NucI. Phys. B361 (1991) 93
[21] J. Sehwinger, J. Math. Phys. 2 (1961) 407 [22] L.V. Keldysh, ZhETF 47 (1964) 1515
123] K. Chou et al., Phys. Rep. 118 (1985) 1;
NP. Landsman and Ch.G. van Weert, Phys. Rep. 145 (1987) 141 [24] V.V. Khoze and A. Ringwald. CERN preprint CERN-TH-6082/91 (1991) [25] V.V. Khoze and A. Ringwald, Phys. Lett. B259 (1991) 106
[26] D. Foerster, Phys. Lett. B66 (1977) 279 [27] E.B. Bogomolny, Phys. Lett. B91 (1980) 431;
J. Zinn Justin, NucI. Phys. B192 (1981) 125
[28] A.H. Mueller. NucI. Phys. B348 (1991) 310: B353 (1991) 44 [29] H. Levine and L. Jaffe, Phys. Rev. D19 (1979) 1225
1301 5. Yu, Khlebnikov and PG. Tinyakov. Phys. Lett. B269 (1991) 149 [31] N. Andrei and D.J. Gross, Phys. Rev. D18 (1978) 468
[32] PB. Arnold and M.P. Maths, Los Alamos preprint, LA-UR-9l-1858 (1991)